Critical behavior of the random-field Ising model with long-range interactions in one dimension

Abstract

We study the critical behavior of the one-dimensional random field Ising model (RFIM) with long-range interactions ($\propto r^{-(d+\sigma )}$) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behavior as a function of $\sigma$, separated by a critical value ${\sigma }_c$. What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behavior can be associated to the characteristics of the large-scale avalanches present in the system at zero temperature. We propose ways to check these predictions through lattice simulations. We also discuss the difference with the RFIM on the Dyson hierarchical lattice.

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• Dimensional reduction breakdown and correction to scaling in the random-field Ising model
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• One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O($N$) spin model near lower critical dimension
• Existence of long-range order in random-field Ising model on Dyson hierarchical lattice
• On the breakdown of dimensional reduction and supersymmetry in random-field models
• The Schwartz-Soffer and more inequalities for random fields
• One-dimensional long-range Ising model: two (almost) equivalent approximations